若数列 xn{x_n}xn yn{y_n}yn zn{z_n}zn 从某项起满足 yn≤xn≤zny_n \leq x_n \leq z_nyn≤xn≤zn 且 yn{y_n}yn zn{z_n}zn 极限为a 则 {xn}\{x_n\}{xn} 的极限存在, 且 limn→∞xn=a\displaystyle\lim _{n \to \infty} x_n=an→∞limxn=a
以上定理可以推广到函数的极限: 若 x∈U˚(x0,r)x\in \mathring{U}(x_{0},r)x∈U˚(x0,r)或∣x∣>M|x|>M∣x∣>M时, g(x)≤f(x)≤h(x)g(x)\leq f(x)\leq h(x)g(x)≤f(x)≤h(x) 且 limx→x0/∞g(x)=limx→x0/∞h(x)=A\displaystyle\lim_{ x \to x_{0}/\infty }g(x)=\lim_{ x \to x_{0}/\infty }h(x)=Ax→x0/∞limg(x)=x→x0/∞limh(x)=A 则limx→x0/∞f(x)=A\displaystyle \lim_{ x \to x_{0}/\infty }f(x)=Ax→x0/∞limf(x)=A , 极限存在
